Lusin characterisation of Hardy spaces associated with Hermite operators

Abstract: Let $d \in {3, 4, 5, \ldots}$ and $p \in (0,1]$. We consider the Hermite operator $L = -\Delta + |x|^2$ on its maximal domain in $L^2(\mathbb{R}^d)$. Let $H_L^p(\mathbb{R}^d)$ be the completion of $ { f \in L^2(\mathbb{R}^d): \mathcal{M}L f \in L^p(\mathbb{R}^d) } $ with respect to the quasi-norm $ |\cdot|{H_L^p} = |\mathcal{M}\cdot|{L^p}, $ where $\mathcal{M}_L f(\cdot) = \sup{t > 0} |e^{-tL} f(\cdot)|$ for all $f \in L^2(\mathbb{R}^d)$. We characterise $H_L^p(\mathbb{R}^d)$ in terms of Lusin integrals associated with Hermite operator.